Hannan Limitation for 3D Antenna Array

Hannan Limitation provides the radiation efficiency limit for large 2D planar antenna arrays, which demonstrates the inevitable radiation efficiency degradation caused by mutual coupling effects between array elements. However, this limitation is derived based on the assumption of infinitely large 2D arrays, which means that it is not an accurate law for small-size arrays. In this paper, we extend this theory and propose an estimation formula for the upper limit of radiation efficiency for finite-sized 2D arrays. Furthermore, we derive the fundamental efficiency and gain limit for a double-layer 3D array structure. Through simulations, we further demonstrate the effectiveness of our proposed theories and gain advantages of 3D arrays under various scenarios and conditions.